Pay for it since you can with no issues. College costs are so much higher relative to our day that it's not practical for a kid to pay for an expensive private college him/herself. Work study and summer work will barely cover the meal plan. And the fact that you have high income means that they won't qualify for financial aid so it seems unfair to then stick them with the bill. Where do you expect them to get $250,000 of loans anyway?
Frankly you should have had this conversation before they applied to college. If you plan to go through with it I hope they've also applied to a state school, or lesser schools where they could qualify for merit aid, or have the option of deferring and maybe emancipating so they can claim financial aid going forward.